Diagonal harmonics, Hopf algebras, and Polytopes

The starting point for this project is a surprising relationship that we recently discovered between diagonal harmonics, Hopf algebras, and polytope theory. Studying the unexpected connections between these seemingly disparate fields requires a broad range of expertise across different areas, including algebraic combinatorics, discrete geometry, algebra, representation theory, and symmetric functions. We propose to forge novel connections between these areas by attacking a selection of open problems relating them. Our trilateral approach will open new, unexplored avenues for future research. Our analysis will require the implementation of algorithms that will contribute to the development of free open source software.

Mathematics often advances when ideas from seemingly different areas are brought together. The project "Diagonal harmonics, Hopf algebras, and Polytopes" explored such connections between algebra, combinatorics and geometry. Its central aim was to understand complicated mathematical objects by finding simpler structures that can describe them from several different perspectives. One major achievement was the development of new connections between algebraic structures known as Hopf algebras and diagonal harmonics, an area at the heart of several important problems in modern combinatorics. Together with collaborators, we introduced new combinatorial objects called Hopf chains and showed that they can be used to describe important algebraic structures in concrete combinatorial terms. This provides a new framework for studying multivariate diagonal harmonics. The long-term goal of obtaining a complete understanding of these structures remains an active area of research. A second major direction concerned two of the most important objects in combinatorial geometry: the permutahedron and the associahedron. These are high-dimensional geometric shapes whose vertices and edges encode rich mathematical structures. We introduced broad generalizations of these objects and established their underlying combinatorial and geometric properties. In particular, we developed a unified framework connecting several families of generalized Tamari lattices and associahedra. This work shows that structures which at first appear unrelated can in fact be different manifestations of the same underlying principles. The project also led to important advances in the study of pipe dreams and related combinatorial structures. We established a natural lattice structure on acyclic pipe dreams and connected it to classical ordering structures on permutations. Further results in the broader geometric setting of subword complexes created new links between combinatorics, geometry and algebra. An important direction that emerged during the project was the study of flow polytopes, geometric objects arising naturally from networks. We developed a new framework, called framing lattices, that connects triangulations of flow polytopes with well-known combinatorial structures such as the Tamari lattice and the weak order. This provides a common language for studying a wide range of seemingly different examples and opens new directions connecting polyhedral geometry, combinatorics and representation theory. Overall, the project resulted in a substantial body of research, including publications in international mathematical journals and new preprints. It also established a long-term collaboration with researchers at York University in Toronto, leading to joint publications and continued research beyond the lifetime of the project. The project demonstrated the value of combining different mathematical viewpoints. Moving between algebraic, combinatorial and geometric perspectives revealed hidden connections, provided new tools for difficult problems, and led to new mathematical structures. Several questions opened during the project remain active areas of research, providing a foundation for future developments.